The Sum of a series with exponents

[islandboy401]

Homework Statement

Problem: Indicate whether the series converges or diverges. If it converges, find its sum.

THE SERIES:

Homework Equations

The ratio test and w/e equation is used to find the sum of this particular series

The Attempt at a Solution

I was able to find that the series converges, using the ratio test. However, I cannot find the sum of the series. I do not see any way in which I could manipulate the geometric series, or anything like that. Could someone please enlighten me on how to find the sum of this series, or any series in such a form.

 

[Dick]

I don't think you need to do a lot of manipulation on a geometric series to find the sum. It should be pretty straightforward. Just factor out the k=1 term.

 

[islandboy401]

I have factored out the k=1 term...yet, I still do not know exactly what to do. I wrote out the series for both the numerator and the denominator, yet I do not see what to do next.

Here is my work:

Another hint at the problem solving process will be greatly appreciated.

Thanks.

 

[rock.freak667]

you could just write 2^k+1=2^k*2 and 5^k-1=5^k*5^-1

 

[islandboy401]

Thanks for all the hints...but now, I have another problem. I obtained an answer of 50/3...however, the solution booklet says the answer is 20/3...

Here is my work:

Is this a typo in the manual, or is my answer truly wrong? If so, please tell me where I messed up.

 

[gabbagabbahey]

The problem is that your sum is going from 1 to infinity, not zero to infinity. The formula for the geometric series that you used requires that the sum goes from zero to infinity:

\sum_{k=0}^{\infty}ax^k=\frac{a}{1-x}

But

\sum_{k=1}^{\infty}ax^k=\frac{a}{1-x}-a